A PTAS for $k$-hop MST on the Euclidean plane: Improving Dependency on $k$

Abstract

For any $\epsilon>0$, Laue and Matijevi\'{c} [CCCG'07, IPL'08] give a PTAS for finding a $(1+\epsilon)$-approximate solution to the $k$-hop MST problem in the Euclidean plane that runs in time $(n/\epsilon)^{O(k/\epsilon)}$. In this paper, we present an algorithm that runs in time $(n/\epsilon)^{O(\log k \cdot(1/\epsilon)^2\cdot\log^2(1/\epsilon))}$. This gives an improvement on the dependency on $k$ on the exponent, while having a worse dependency on $\epsilon$. As in Laue and Matijevi\'{c}, we follow the framework introduced by Arora for Euclidean TSP. Our key ingredients include exponential distance scaling and compression of dynamic programming state tables.